Most problems in physics just cannot be determined exactly. I often talk to people who cannot imagine what that means and how this effect arises. To make things a little bit clearer, let us try to understand how physicists describe the conductance of solids. Bear with me and you will get an impression of the physicists’ profession.
Some good news right away: The formula to describe all solids in the universe is known. This formula has already been known for quite some time, more than 100 years to be exact. Unfortunately, there is no way to solve it.
This is quite a strong statement and to understand it, it is important to first understand the equation itself. Solids consist of atoms, which are again just positively charged cores, also called ions, surrounded by negatively charged electrons. For the sake of this article, think of them as marbles. They interact via electrical force, also called the Coulomb force. The Coulomb force depends only on the distance of the interacting particles and their charge. An electron and an ion attract each other, and two electrons or ions interact repulsively. Finally, it is necessary to know the particles’ movement, aka, their momentum. Only the information of the forces acting on a particle, and its momentum, allows for predictions of its trajectory.
So far, so abstract. Let’s try to make an intuitive example. Consider a golf ball on a hilly mini-golf course. To predict where the ball will be a second from now, you need to know the hill’s steepness (potential/interaction), as well as its current velocity (momentum).
In this problem, however, there are millions and millions of golf balls on an ever-changing golf course. To solve the problem, it would be necessary to solve one equation for each particle, which each depends on all the others. That is a task no human will ever be able to tackle.
Okay, so what? There are computers these days, just let them do the work. Unfortunately, that’s not possible either. In a typical object in our world, there are just too many atoms. As a rule of thumb, there are about 10^23 atoms in one cubic centimetre of matter. Writing this number out, we get 100.000.000.000.000.000.000.000. To take this into scale, look at modern computers. A modern computer can do 10^9 calculations per second (the clock frequency of modern CPUs is in the GHz regime). That means calculating the behaviour of a bottle of water would take the age of the universe.
And even if it was possible to solve this exactly, the solution would be useless. The initial positions and momenta of the particles would have to be plugged into the solution and getting all this information just is impossible.
What a tough start! And quite a big disclaimer. Nevertheless, the electrical behaviour of solids seems to be something that is very well understood. So, what did physicists come up with to work around this unsolvable problem?
The first step is to get rid of the idea of solving things exactly. Rather the physicists’ favourite tool comes into play: approximations. They are in use in every single area of physics, since most of the time, the world is way too complicated. Instead, the approach is to identify the scales and effects, responsible for the phenomenon of interest. The electrical behaviour of metals in this case.
The first useful property is the fact that ions are several thousand times heavier than electrons. That means, if they got the same amount of energy, electrons move much faster than ions. Just compare kicking a football to kicking a steel version of it. Which one will be faster? Since the goal is to understand electrical properties, electrons are the stars of the show, and from their perspective, ions only barely move. Therefore, assuming the ions to be fixed should not mess with the results too much. Just like this, there are 10^23 equations less. A big win, but the problem is still far too complex to be solved exactly.
The next simplification does not arise from an approximation. Clever calculations reveal that these fixed ions order in periodic structures or lattices. As mentioned above, ions repel other ions. If they are forced into a certain volume, the best way to reduce the repelling force between the ions is to all have the same distance from each other. Imagine a chequered piece of paper with one ion per intersection. In complex solids, the lattices are more complex as well. Sodium chloride, e.g., also known as table salt, has four ions at each intersection.
This periodicity means every few atoms, the problem repeats. What is left is the interaction of just a few ions with all the electrons. It seems like not much has been achieved. But one last approximation saves the day. Remember the Coulomb interaction? This force quickly decays with growing distance. So why not neglect particles far away and only include the interactions with the ones nearby? To make our lives as simple as possible, just consider the very next ion of each electron.
Finally, the problem is solved by the interactions of a few ions with their electrons. And that’s why physicists like approximations so much! It took three assumptions to break an unsolvable problem down to a model that could be calculated by hand. This theory is can describe the behaviour of metals, insulators, semiconductors, and even strange quantum phenomena, such as superconductivity, but this is a story for a different time.
4 replies on “The unsolvable problem”
Can we do this only for materials that are ordered in lattices? Or are there even solids with atoms that are not ordered in lattices? I imagine if such materials exist, you couldn’t make that simplification?
Yes, we can only do this for materials that order in lattices. Fortunately, most materials consisting of only a few different atoms do so (organic materials, polymers etc. are too complicated to be described this way). However, some simple materials don’t. In fact, there are so many, that they got their own name: glasses. Why they occur and how they can be described is still subject to ongoing research.
I have heard once that a glass should actually be considered a fluid and not a solid. Does this have anything to do with their atoms not arranging in lattices just like, as I assume, other fluids?
The short answer is yes. The more in-depth one goes like this: Under certain circumstances, you can cool down a liquid below its freezing temperature without it freezing. This is called supercooling.
How does it work? Upon cooling down, molecules move slower. If, for example, a liquid is cooled down fast enough, the molecules can only move slow before they have the chance to order in a lattice. Such a state is called metastable. The liquid wants to become a solid, but some energy is needed to start the process.
This is only a very brief summary! The underlying concept is that of phase transitions. If you are keen to know more, I found a nature article that summarizes this topic. It might be a bit too technical, but the first few paragraphs nicely explain what glasses are about.